You can find the correlation coefficient between two variables by taking the square root of the R-squared value (R2) of a simple linear regression model.
Correlation coefficient = √R2 of simple linear regression model
The sign of the slope coefficient in the regression model tells you whether the correlation coefficient is positive or negative.
The following examples show how to find the correlation coefficient from the R-squared value of a regression model in practice.
Note: The R-squared value of a regression model is also called the coefficient of determination.
Example 1: Find Correlation Coefficient from R2 (when slope is positive)
Suppose we fit a simple linear regression model using hours studied as the predictor variable and exam score as the response variable.
Suppose we receive the following output from the model:
Fitted Regression Equation: Exam Score = 65.55 + 2.78(Hours Studied)
R-Squared (R2) of Regression Model: 0.7845
The R-squared value of the model tells us what percentage of variation in exam scores can be explained by hours studied.
In this example, we can see that hours studied is able to explain 78.45% of the variation in exam scores.
To find the correlation coefficient between hours studied and exam scores, we can take the square root of R2:
Correlation coefficient = √R2 = √0.7845 = 0.8857
Since the sign is positive for hours studied in the regression equation, this correlation coefficient is positive.
Thus, the correlation coefficient between hours studied and exam score is 0.8857.
Example 2: Find Correlation Coefficient from R2 (when slope is negative)
Suppose we fit a simple linear regression model using age (in years) as the predictor variable and max bench press (in pounds) as the response variable.
Suppose we receive the following output from the model:
Fitted Regression Equation: Max bench press = 240.11 – 1.24(Age)
R-Squared (R2) of Regression Model: 0.4773
The R-squared value of the model tells us what percentage of variation in max bench press pounds can be explained by age.
In this example, we can see that age is able to explain 47.73% of the variation in max bench press amount.
To find the correlation coefficient between age and max bench press, we can take the square root of R2:
Correlation coefficient = √R2 = √0.4773 = 0.6909
Since the sign is negative for age in the regression equation, this correlation coefficient is negative.
Thus, the correlation coefficient between age and max bench press is -0.6909.
Additional Resources
The following tutorials provide additional information about correlation coefficients:
What is Considered to Be a “Strong” Correlation?
When Should You Use Correlation?
How to Perform a t-Test for Correlation
